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A connected graph is a graph in which we can get from any vertex to
any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops. Prove that every tree has
exactly one more vertex than it has edges.

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

Olympic Magic

Stage: 4 Challenge Level:

Thank you to Alma Iulia Dimitriu, of St
Peter's RC High School, and member of the NRICH/NAGTY mentoring
scheme), for this solution.

We are working here with vertex magic graphs.

When we draw a graph for the problem there are 5 vertices and 4
edges. A vertex sum is the sum of the numbers on the vertex and on
the edges joined to that vertex. The sum in each circle (which can
be either 10,11,12,13,14 or15) is the vertex sum in the graph and
we want to make this a constant X to make it a magic sum.

The sum of all the numbers is: 1+2+3+4+5+6+7+8+9 = 45

The edges are used twice. There are 4 numbers included in two
circles and the sum of the 4 edges that are used twice is
a+b+c+d=n, so n + 45 = 5X. So n has to be multiple of 5.

Also 1+2+3+4 = 10 so X must be greater than or equal to 10. So
which means that 10 cannot be the magic sum.

For X to be 15, n has to be 30 and the only 4 numbers that add
up to 30 are 6+7+8+9. It is impossible for any of them to be on the
edges because the only pairs of numbers that add up to 15 are 9+6
and 7+8 and two of these numbers have to be placed at the two outer
vertices.

For X to be 12, n has to be 15. There can be 3 pairs of numbers
that equal 15, that is 10+5; 9+6 and 8+7. There are 4 numbers to
make n because there are 4 edges. So all the possible choices of
the 4 numbers on the edges are:

Table 1: Possible choices of the 4
numbers

For 10+5

For 9+6

For 8+7

1+9+2+3

1+8+2+4

1+7+2+5

2+8+4+1

2+7+1+5

1+7+3+4

3+7+4+1

3+6+2+4

2+6+3+4

4+6+2+3

3+6+1+5

3+5+1+6

The only two numbers that equal 12 (for the 2 circles on the
margins) are: 9+3; 8+4 and 7+5.

So v1 and v5 have to be any two of those numbers. So now the
possibilities left for the edges are:

For 10+5:

1+9+2+3 - no solution as the only two numbers here are 3+9=12
(they can't be separated)
2+8+4+1 - no solution the same as above
3+7+4+1 - a possible solution because for v1=7 and v5=4
4+6+2+3 - no solution

For 9+6: - no solutions

For 8+7: 1+7+3+4 (the same as the one above)

So now we only have two possible solutions where 7 and 4 have to
be on the margins:

Case (i) a=4, b=1, c=3, d=7
a=4 needs v1=8 to make 12. The second circle already has 4 and 1 so
it will need v2=7 to make 12. This cannot be a solution because we
already have d=7.

Case (ii) a=7, b=1, c=3, d=4
a=7 will need v1=5 to make 12. The second circle has 7 and 1 and it
will need v2=4 but this is impossible as d=4.

So there is no possible solution with X equal to10, 12 or 15

The remaining values for X are 11, 13 and 14 for which 5X=55, 65
and 70. As n + 45 = 5X we have n=10, 20 and 25.

For X=11, the only numbers for the edges to add up to 10 are:
1+2+3+4 and the only pairs to make 11 are: 9+2, 8+3, 7+4 and 6+5.
Checking all the possible cases gives the following solution and
its mirror image:

v1=8, a=3, v2=7, b=1, v3=6, c=4, v4=5, d=2, v5=9

For X=13 we have a+b+c+d=n=20 and checking all possible cases
gives two solutions and their mirror images;

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